In logic and mathematics – especially metalogic and computability theory – an effective method (also called an effective procedure) is a procedure which takes some class of problems and reduces the solution to a set of steps which:
- always give some answer rather than ever give no answer;
- always give the right answer and never give a wrong answer;
- always be completed in a finite number of steps, rather than in an infinite number;
- work for all instances of problems of the class.
An effective method for calculating the values of a function is an algorithm; functions with an effective method are sometimes called effectively calculable.
Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions (general recursion, Turing machines, λ-calculus) that later were shown to be equivalent; the notion captured by these definitions is known as (recursive) computability.
The Church–Turing thesis states that the two notions coincide: any number-theoretic function that is effectively calculable is recursively computable. This is not a mathematical statement and cannot be proven by a mathematical proof.
A further elucidation of the term "effective method" may include the requirement that, when given a problem from outside the class for which the method is effective, the method may halt or loop forever without halting, but must not return a result as if it were the answer to the problem.
An essential feature of an effective method is that it does not require any ingenuity from any person or machine executing it.
Famous quotes containing the words effective and/or method:
“The only effective way to help well-intentioned, intelligent persons to do the best they can in raising children is to encourage and guide them always to do their own thinking in their attempts at understanding and dealing with child-rearing situations and problems, and not to rely blindly on the opinions of others.”
—Bruno Bettelheim (20th century)
“The good husband finds method as efficient in the packing of fire-wood in a shed, or in the harvesting of fruits in the cellar, as in Peninsular campaigns or the files of the Department of State.”
—Ralph Waldo Emerson (1803–1882)